Are the two affine varieties $V(x_2^2-x_1^3-x_1^2)$ and $V(x_1^2-x_2^2-1)$ isomorphic over $\mathbb{C}$?

Question

This is a part of Exercise 4.19 of Gathmann's 2021 notes of Algebraic Geometry.

For $V(x_2^2-x_1^3-x_1^2)$, we have $x_2^2=x_1^2(x_1+1)$ and $ x_2^2-x_1^3-x_1^2$ is irreducible by Eisenstein's criterion.

For $V(x_1^2-x_2^2-1)$, we have $x^2_2=(x_1-1)(x_1+1)$ and $x_1^2-x_2^2-1$ is also irreducible by Eisenstein's criterion.

Therefore their corresponding rings of regular functions are both not a UFD and are both integral domains.

So it seems not easy to show that these two are not isomorphic by showing that their corresponding rings of regular functions are not isomorphic.

Conversely, I just cannot come up with any reasonable morphism between these two that may be an isomorphism...

Thanks in advance for any help.

Answer 1

$V(x_2^2-x_1^3-x_1^2)$ is singular at $(0,0)$, while $V(x_1^2-x_2^2-1)$ has no singular points.